Hi there, welcome!
As a small tip, it might be easier to read your post with the normal font.
As for the meter, there are experts here that might be able to help.
As for the symptoms, as you yourself describe them as strange sensations, it could be placebo as you say, that is, perhaps you expect something to go wrong (maybe you don't feel the meter is very good). Maybe once you have a little bit more confidence, it will feel better? As a newbie myself, when working on projects sometimes I feel "this wire/resistor/transistor/etc must be getting hot", because I almost expect to have shorted something or for something to go wrong.
But usually when I touch it again, I notice it's not hot at all, unless if I did really short something.
As for the safety of batteries, I'm by no means an expert, but I do believe the batteries are not unsafe when used properly. Perhaps when shorting them out (e.g. with a piece of wire), you can get very high temperatures, and burn your finger if you're holding the wire. Make sure to check the voltage, and not current of batteries (or any other power source) directly with a multimeter.